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Aim: What is a function & what are the different ways to represent them? Do Now: Graph the function y = x2 using a table. Be sure to include the interval -2 ≤ x ≤ 2. A relation is a set of ordered pairs. The domain is the set of all x values in the relation Domain: {-1,0,2,4,9} These are the x values written in a set from smallest to largest {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} These are the y values written in a set from smallest to largest Range: {-6,-2,3,5,9} The range is the set of all y values in the relation This is a relation A relation assigns the x’s with y’s 1 2 3 4 2 4 6 5 8 10 Domain (set of all x’s) Range (set of all y’s) This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)} AAfunction function ffisisaarelation relation that thatassigns assigns to toeach each element element xxin in the thedomain domain to toexactly toA exactly exactlyone oneelement elementy yininthe therange. range.. 1 2 3 4 5 Set A is the domain 2 4 6 8 10 Set B is the range Must use all the x’s This is a function ---it meets our The x value can only be assigned to one y conditions Let’s look at another relation and decide if it is a function. The second condition says each x can have only one y, but it CAN be the same y as another x gets assigned to. 1 2 3 4 5 2 4 6 8 10 Set A is the domain This is a function ---it meets our conditions Set B is the range Must use all the x’s The x value can only be assigned to one y Relating functions to real life… Imagine Set A is the set of students in the class, and the grades they can earn is set B. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). David A B C D F Samantha Sahid Angela Alfredo Is the relation shown above a function? NO Why not??? Check this relation out to determine if it is a function. It is not---3 didn’t get assigned to anything Comparing to our example, a student in math must receive a grade 1 2 3 4 5 Set A is the domain 2 4 6 8 10 Set B is the range This is not a Must use all the x’s function---it doesn’t assign each x with a y The x value can only be assigned to one y Check this relation out to determine if it is a function. This is fine—each student gets only one grade. More than one can get an A and I don’t have to give any D’s (so all y’s don’t need to be used). 1 2 3 4 5 2 4 6 8 10 Set A is the domain This is a function Set B is the range Must use all the x’s The x value can only be assigned to one y Determine whether each relation is a function. 1. {(2, 3), (3, 0), (5, 2), (4, 3)} 2 3 5 4 f(x) f(x) f(x) f(x) 3 0 2 3 YES, every domain is different! Determine whether the relation is a function. 2. 4 5 5 6 1 {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} f(x) f(x) f(x) f(x) f(x) 1 2 3 6 9 NO, 5 is paired with 2 numbers! Is this relation a function? {(1,3), (2,3), (3,3)} 1. Yes 2. No Answer Now Practice Practice Vertical Line Test If any vertical line passes through more than one point of the graph, then that relation is not a function. Are these functions? FUNCTION! FUNCTION! NOPE! Vertical Line Test FUNCTION! NO! NO WAY! FUNCTION! Is this the graph of a function? 1. Yes 2. No Answer Now Aim: How do we find the value of a function? Do Now: State whether the following relations are also functions & explain why: a. { (-3, 4), (-2, 4), (-1, 4), (0, 4) } b. { (-2, 1), (-2, 2), (-2, 3), (-2, 4) } We commonly call functions by letters. Because function starts with f, it is a commonly used letter to refer to functions. f x 2 x 3x 6 2 This means the right hand side is a function called f This means the right hand side has the variable x in it The left side DOES NOT MEAN f times x like brackets usually do, it simply tells us what is on the right hand side. The left hand side of this equation is the function notation. It tells us two things. We called the function f and the variable in the function is x. Remember---this tells you what is on the right hand side---it is not something you work. It says that the right hand side is the function f and it has x in it. f x 2 x 3x 6 2 f 2 22 32 6 2 f 2 24 32 6 8 6 6 8 So we have a function called f that has the variable x in it. Using function notation we could then ask the following: This means to find the function f and instead of having an x in it, put a 2 in it. So let’s take the Find f (2). function above and make brackets everywhere the x was and in its place, put in a 2. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction Find f (-2). f x 2 x 3x 6 2 f 2 2 2 3 2 6 2 f 2 24 3 2 6 8 6 6 20 This means to find the function f and instead of having an x in it, put a -2 in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a -2. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction f x 2 x 3x 6 2 Find f (k). f k 2k 3k 6 2 f k 2 k 3k 6 2k 3k 6 2 2 This means to find the function f and instead of having an x in it, put a k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a k. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction f x 2 x 3x 6 2 Find f (2k). f 2k 22k 32k 6 2 f 2k 2 4k 32k 6 8k 6k 6 2 2 This means to find the function f and instead of having an x in it, put a 2k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a 2k. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction Let's try a new function Find g(1)+ g(-4). g x x 2 x 2 g 1 1 21 1 2 g 4 4 2 4 16 8 24 2 So g 1 g 4 1 24 23 The last thing we need to learn about functions for this section is something about their domain. Recall domain meant "Set A" which is the set of values you plug in for x. For the functions we will be dealing with, there are two "illegal” situations: 1. You can't divide by zero (denominator (bottom) of a fraction can't be zero) 2. You can't take the square root (or even root) of a negative number When you are asked to find the domain of a function, you can use any value for x as long as the value won't create an "illegal" situation. Find the domain for the following functions: Since no matter what value you choose for x, you won't be dividing f x 2x 1 by zero or square rooting a negative number, you can use anything you Note: There is want so we say the answer is: nothing wrong with the top = 0 just means All real numbers x. the fraction = 0 x3 g x x2 illegal if this is zero If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is: All real numbers x such that x ≠ 2. means does not equal Let's find the domain of another one: h x x 4 Can't be negative so must be ≥ 0 x4 0 solve this x4 We have to be careful what x's we use so that the second "illegal" (taking the square root of a negative) doesn't happen. This means the "stuff" under the square root must be greater than or equal to zero (the mathematical way of saying "not negative"). So the answer is: All real numbers x such that x ≥ 4 Summary of How to Find the Domain of a Function • Look for any fractions or square roots that could cause one of the two "illegals" to happen. If there aren't any, then the domain is All real numbers x. • If there are fractions, figure out what values would make the bottom equal zero and those are the values you can't use. The answer would be: All real numbers x such that x ≠ those values. • If there is a square root, the "stuff" under the square root cannot be negative so set the stuff ≥ 0 and solve. Then answer would be: All real numbers x such that x ≠ whatever you got when you solved. NOTE: Of course your variable doesn't have to be x, can be whatever is in the problem.